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The ground state of nanoscale circular magnetic disks of certain geometric aspect ratios is a spontaneously forming stable vortex configuration with circulating in-plane magnetization and a vortex core pointing out-of-plane. Resonantly exciting the vortex core via either an rf magnetic field or an rf spin-polarized current yields a gyrotropic motion around its equilibrium position, characterised by a specific eigen-frequency, which depends on the material parameters and the disk geometry [1]. Such oscillations, which can be read out via periodic magnetoresistance changes, generate rf signals with high quality factors (&gt;10000) in the sub-GHz bandwidth [2,3]. While all these features make vortex-based nano-oscillators interesting as nanoscale rf sources, the major drawback remains their low frequency tunability associated with the linear characteristics of the gyrotropic mode. Here, we investigate, by micromagnetic simulations, the role of magnetostriction in improving the tunability of vortex nano-oscillators. Specifically, we consider a double-disk structure comprising magnetostrictive (CoFe) and non-magnetostrictive (Py) layers separated vertically by a non-magnetic (NM) spacer. We show that, when the two vortices have different eigen-frequencies and the magnetostatic coupling between them is sufficiently strong, the stress-induced magnetoelastic anisotropy can lead to the synchronized gyration of the two vortex cores (Fig. 1). The stress-induced transition from double-frequency to single-frequency dynamics is mostly controlled by the polarization of the vortices and the magnetostatic coupling strength (i.e. spacer thickness). These findings offer a frequency tunability of vortex-based oscillators via mechanical stress, which can be generated and controlled electrically, for example, using piezoelectric substrates [4]. Funding from the EU Horizon 2020 project No. 737038 (TRANSPIRE) is acknowledged.References: [1] K. Yu. Guslienko, et al, J. Appl. Phys. 91, 8037 (2002). [2] A. Dussaux, et al, Nat. Commun. 1, 1 (2010). [3] N. Locatelli, et al, Appl. Phys. Lett. 98, 062501 (2011). [4] M. Filianina, et al, Appl. Phys. Lett. 115, 062404 (2019).

MMM 2020

Stress-Induced Modification of Gyration Eigen-Frequencies in Stacked Double-Vortex Structures

The 65th Annual Conference on Magnetism and Magnetic Materials (MMM 2020) was held in a virtual form. The COVID-19 pandemic, with its associated uncertainty about international travel and the ability to gather in large groups, meant that the Steering Committee decided to cancel the face-to-face meeting that had been planned in Palm Beach, Florida November 2-6, 2020.

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The ground state of nanoscale circular magnetic disks of certain geometric aspect ratios is a spontaneously forming stable vortex configuration with circulating in-plane magnetization and a vortex core pointing out-of-plane. Resonantly exciting the vortex core via either an rf magnetic field or an rf spin-polarized current yields a gyrotropic motion around its equilibrium position, characterised by a specific eigen-frequency, which depends on the material parameters and the disk geometry [1]. Such oscillations, which can be read out via periodic magnetoresistance changes, generate rf signals with high quality factors (>10000) in the sub-GHz bandwidth [2,3]. While all these features make vortex-based nano-oscillators interesting as nanoscale rf sources, the major drawback remains their low frequency tunability associated with the linear characteristics of the gyrotropic mode. Here, we investigate, by micromagnetic simulations, the role of magnetostriction in improving the tunability of vortex nano-oscillators. Specifically, we consider a double-disk structure comprising magnetostrictive (CoFe) and non-magnetostrictive (Py) layers separated vertically by a non-magnetic (NM) spacer. We show that, when the two vortices have different eigen-frequencies and the magnetostatic coupling between them is sufficiently strong, the stress-induced magnetoelastic anisotropy can lead to the synchronized gyration of the two vortex cores (Fig. 1). The stress-induced transition from double-frequency to single-frequency dynamics is mostly controlled by the polarization of the vortices and the magnetostatic coupling strength (i.e. spacer thickness). These findings offer a frequency tunability of vortex-based oscillators via mechanical stress, which can be generated and controlled electrically, for example, using piezoelectric substrates [4]. Funding from the EU Horizon 2020 project No. 737038 (TRANSPIRE) is acknowledged.References: [1] K. Yu. Guslienko, et al, J. Appl. Phys. 91, 8037 (2002). [2] A. Dussaux, et al, Nat. Commun. 1, 1 (2010). [3] N. Locatelli, et al, Appl. Phys. Lett. 98, 062501 (2011). [4] M. Filianina, et al, Appl. Phys. Lett. 115, 062404 (2019).

poster

Spin orbit torques (SOTs) are being actively considered for use in the next generation memory devices for magnetization switching, spin oscillators and racetrack memories, including those using magnetic skyrmions. SOTs are fundamentally based on charge-to-spin conversion by the spin-Hall or Rashba effect. Taking advantage of the spin current, the torques can be used to switch magnetic insulators or the free layer of magnetic tunnel junctions with high energy efficiency. Therefore, it is important to accurately measure the charge-to-spin conversion efficiency. SOTs are principally of interest in nanostructured samples, samples with minimum dimension less than a micron. However, the magnitude and form of the torques are most often characterized in micron scale samples, using spin torque ferromagnetic resonance (ST-FMR), a technique that involves analyzing the resonance linewidth [1] or lineshape [2]. On microstructures, these two analysis methods are quite consistent. Here we present ST-FMR results on permalloy Ni80Fe20 nanowires --- with widths varying from 150 to 800 nm --- that show that the standard model used to analyze the resonance linewidth and lineshape give different results [3]. Fig. 1 a-c show examples of ST-FMR spectra. Fig. 1d shows the efficiency of nanowires with different width, which shows greatly enhanced efficiency when the lineshape method is used. We present a ST-FMR model that properly accounts for the sample shape. It shows a much better consistency between the two methods, as illustrated in Fig. 2. Micromagnetic simulations are used to verify the model. These results and the more accurate nanowire model presented are of importance for characterizing and optimizing charge-to-spin conversion efficiencies in nanostructures.References: [1] S. Petit et al., Physical Review Letters 98, 077203 (2007) [2] L. Liu et al., Physics Review Letters 106, 036601 (2011). [3] J.-W. Xu et al., arXiv:2004.03784(2020). [to appear in Physical Review Applied]

Charge-to-spin conversion efficiency in ferromagnetic nanowires by spin torque ferromagnetic resonance: Reconciling lineshape and linewidth analysis methods

Generation and manipulation of pure spin current play an important role in fabrication of power efficient spintronics devices1. Spin pumping is very efficient way of creation of pure spin current in ferromagnetic (FM)/nonmagnetic (NM) system by microwave excitation which can be converted into a voltage by inverse spin Hall effect (ISHE) Various materials possessing high spin orbit coupling (SOC) like Pt, Ta, W, Bi2Se3 etc., have been utilized for efficient spin to charge conversion2,3. However, these are limited by very small spin diffusion length (λsd ~10 nm). In this context organic semiconductors (OSC) e.g. C60 are emerging as a potential candidate due to large value of λsd. The low value of SOC in OSC limits them for efficient spin to charge conversion. However, when the OSC is deposited on a substrate the structural distortion may exhibit enhancement of SOC. Here we report simultaneous observation of spin pumping and ISHE measurements in the CoFeB/C60 bilayers by ferromagnetic resonance (FMR) in the frequency range 6 – 17 GHz. During the measurement the sample was kept on the wide coplanar waveguide (CPW) in the flip-chip manner. Angle dependent ISHE measurements have been performed at fixed frequency of 7 GHz and 11 mW power by using a nanovoltmeter at the two edges of the sample. Figure1 shows dc voltage (Vmeas) and FMR signal versus applied magnetic field (H) plot. At 180-degree ISHE signal is reversed as the magnetic field direction is reversed, which indicates that signal is not coming from any artefacts and it is due to spin pumping. Also, we have calculated the spin Hall angle for CoFeB/C60 bilayer system is 0.04 which is higher than the reported value.References: 1. Y. Tserkovnyak, A. Brataas, et al., Rev. Mod. Phys. 77, 13751421 (2005) 2. B. B. Singh et al., physica status solidi (RRL) Rapid Research Letters 13, 1800492 (2019) 3. B. B. Singh et. al., Phys. Rev. Appl. 13, 044020 (2020)

Observation of inverse spin hall effect at CoFeB/C60 interface

2D-transition metal dichalcogenides (TMD) are known to have high spin-orbit coupling, an essential property for spintronic applications [1,2]. TMD materials with this property can replace the heavy metals for next generation spintronic devices. In this context MoS2 has been of great interest in recent years due to its unique electrical and optoelectronics properties. It can be fabricated in very thin layers due to weak Van der waals forces, which makes it suitable to investigate interface dominated Rashba and Edelstein effects. However, fabrication of MoS2 by chemical or exfoliation methods makes it difficult for spintronics applications. In this work, we demonstrate inverse spin Hall effect in MoS2(t)/CoFeB(10nm)/MgO(2nm) heterostructures where ‘t’ varies from 7 to 30nm.The heterostructure was fabricated by RF and DC sputtering of MoS2 and CoFeB, respectively, on thermally oxidized Si substrate. MgO capping layer was deposited by thermal evaporation. Here the spin current was generated from CoFeB and injected into MoS2 layer via spin pumping using ferromagnetic resonance (FMR). The resulting spin to charge conversion at interface was measured using nanovoltmeter. Various spin rectification effects were separated by performing angle dependent measurements [3]. For the sample with MoS2(30nm), a spin pumping voltage of -1.6 mV was measured which was dominant over the rectification effects. The spin mixing conductance value was found to be ~ 3.72×1018 m-2 which is comparable to that of MoS2 and Pt obtained from literature. Figure 1 shows the measured voltage signal for the sample Si/SiO2/MoS2(30nm)/CoFeB(10nm)/MgO(2nm). In order to understand the origin of this voltage, power and frequency dependent measurements were also performed. The linear increase in the symmetric component of dc voltage with the microwave power further indicated spin pumping [4].References: [1] Qing Hua , Kourosh Kalantar-Zadeh, Andras Kis, Jonathan N. Coleman and Michael S. Strano Nature Nanotechnology, 7, 699–712 (2012) [2] Zehua Hu, Zhangting Wu, Cheng Han, Jun He, Zhenhua Ni and Wei Chen, Chemical Society Reviews, 47, 3100-3128, (2018) [3] Braj B. Singh, Sukanta K. Jena, Manisha Samanta, Kanishka Biswas, Biswarup Satpati and Subhankar Bedanta, Phys. Status Solidi RRL,Volume 13, Issue 3,1800492 (2019) [4] K. Ando, S. Takahashi, J. Ieda, Y. Kajiwara, H. Nakayama, T. Yoshino, K. Harii, Y. Fujikawa, S. Maekawa and E. Saitoh, J. Appl. Phys.109, 103913 (2011).

Inverse Spin Hall Effect in Sputtered MoS2/CoFeB Bilayers

Pure spin currents have been ubiquitous in contemporary spintronics research. Despite its pro- found physical and technological significance, the detection of pure spin current has largely remained indirect, which is usually achieved by probing spin-transfer torque effects or spin-to-charge conversions. By using scanning transmission X-ray microscopy, we report the direct detection and spatial mapping of spin accumulation in a nonmagnetic Cu layer without any direct charge current injection. Such a pure spin current is induced by spin pumping from a NiFe layer and is not accompanied by concomitant charge motion. The observed frequency dependence indicates that the signal is dominated by a coherent, pure spin current, but the magnitude of the spin accumulation suggests also possible additional thermal contributions. Our technique takes advantage of the X-ray magnetic circular dichroism and the synchronization of microwave with X-ray pulses, which together provide a high sensitivity for probing transient magnetic moment. From the detected X-ray signals, we observe two distinct resonance modes induced by spin pumping, which, based on micromagnetic simulations, we attribute to non-linear microwave excitations. Our result provides a new pathway for detecting pure spin currents that originate from many spintronics phenomena, such as spin Hall and spin Seebeck effects, and which can be applied to both metal and insulator spin current sources [1]. Work at Argonne, including experimental design and measurement, sample fabrication, data analysis, and manuscript preparation, was supported by the U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences, Materials Science and Engineering Division under Contract No. DE-AC02-06CH11357. Use of the Stanford Synchrotron Radiation Light source, SLAC National Accelerator Laboratory, is supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences under Contract No. DE-AC02-76SF00515. W.Z. acknowledges the DOE visiting Faculty Program.References: 1.) “Direct Observation of spin accumulation in Cu induced by spin pumping” J. Ding, W. Zhang, B. Jungfleisch, J.E. Pearson, H. Ohldag, V. Novosad, A. Hoffmann, Phys. Rev. Res vol 2(1), p. 013262 (2020)

Direct Observation of Spin Accumulation in Cu Induced by Spin Pumping

This paper presents engineering of magnetic domains and perpendicular magnetic anisotropy (PMA) in magnetic thin films and multilayer (ML) employing ion-beam irradiation [1]. Two model systems have been selected for this study namely, amorphous Tb-Fe-Co (200 nm) film and ultrathin Co/Pd ML (Si (substrate) / Ta (30) / Pd (30) / [Co (3) / Pd (8)] ×50 / Pd (12) Å) which show strong PMA in the pristine state, however, the origin of PMA is different for both the systems. The pristine Tb-Fe-Co film depicts a rectangular hysteresis loop and elongated, inter-connected maze-like magnetic domains with strong out-of-plane contrast. Irradiation at higher fluences (≥1015 ions/cm2) of Ar+ ions at an energy of 200 keV results in transformation of magnetization reversal mechanism from rapid domain wall motion to nucleation and propagation of reverse domains due to the pinning of domain walls. The elongated magnetic domains get fragmented with reduced contrast and interestingly inhomogeneous flux closure patterns are observed at higher fluences, indicating a spin reorientation transition, as shown in Fig. 1 (a). The modifications in magnetic properties are correlated with the structural investigations which reveal the crystallization of Fe-Co phase in the amorphous Tb-Fe-Co matrix at higher energy and fluences of the projectile ions. On the contrary, ion-induced intermixing at the interfaces for perpendicularly magnetized ultrathin Co/Pd ML causes amorphization for the pristine poly-crystalline films. In this case, low energy ion-irradiation (50 keV) does not have significant role in modifying the microstructure, however, control of pinning sites can be achieved through variable fluences which play an important role in the modification of magnetic domains, as depicted in Fig. 1 (b). Micromagnetic simulations on domain modification complement the experimental results.References: 1. J. Fassbender, D. Ravelosona and Y. Samson, Journal of Physics D: Applied Physics 2004, 37 (16), R179-R196.

Engineering Magnetic Domains in Perpendicularly Magnetized Thin Films and Multilayers Using Ion-beam Irradiation

The Dzyaloshinskii-Moriya interaction (DMI) plays a crucial role in the stabilization of chiral spin textures [1]. Interfacial DMI (iDMI) induced by a lack of inversion symmetry and strong spin orbit coupling at the interface of Ferromagnet/Heavy Metal (FM/HM) heterostructures provides various opportunities for future spintronic devices. Recently, iDMI in a thin film of NiFe/Graphene (NiFe/Gr) heterostructures has been detected using Brillouin light scattering [2] and using spin-polarized electron microscopy [3]. In this study, we propose a simple method of measuring DMI by using the Bragg magneto-optical Kerr effect (MOKE). We observe an asymmetric shift in the hysteresis loop, which is proportional to the strength of iDMI [4]. Here, we utilize Bragg-MOKE in longitudinal mode to which occurs when a broad polarized beam is incident on a periodic array of the magnetic features with size comparable to the wavelength of laser. We have investigated the hysteresis loop shift for 1st and 2nd order of diffraction in off-specular geometry for both patterned graphene (Gr/NiFe(10nm)/Ta) and reference (NiFe(10nm)/Ta). We used triangular pattern (Fig. 1) as proposed in Ref [4]. We found a shift in hysteresis loop for the Gr/NiFe/Ta sample (Fig. 2), which likely due to the presence of iDMI that was previously measured using BLS [2]. Hence, the Bragg-MOKE technique can be a potential technique for rapid quantification of iDMI in magnetic heterostructures. We will also present variable field magnetic force microscopy (MFM) and micromagnetic simulations to support these results.References: 1. M. Bode, et al., Nature 447, 190 (2007). 2. A. Kumar, et al. Phys. Rev. B 99.3,035402 (2019). 3. H. Yang, et al. Nat. Mater. 17, 605 (2018). 4. D-S Han, et al. Nano Lett. 16, 4438 (2016).

Evidence of the interfacial Dzyaloshinskii-Moriya interaction in Ni80Fe20/graphene using Bragg-MOKE

The perpendicular magnetic media has the potential to improve data storage beyond 1 Tb/in2. Improving data density is a major challenge in data recording industries due to its sustainability and performance issues [1]. In the present study, by employing Ti as under-layer (UL) and spacer-layer (SL) the bulk perpendicular magnetic anisotropy (PMA) of Tb-Fe thin films is found to be improved considerably. 100 nm thick Tb-Fe film devoid of UL exhibited a maze-like domain pattern with strong out-of-plane (OOP) contrast. With 10 nm thin Ti UL, the domain width, and the OOP contrast of Tb-Fe (100 nm) film were found to improve considerably (fig. 1). Accordingly, the coercive field for Tb-Fe film with Ti UL was doubled when compared with the bare Tb-Fe film. The possible reason for the enhancement in OOP properties could be due to the lateral extent of d-d hybridization that exists at the interface. Similarly, PMA and domain areal density also increased by placing a 2 nm thin Ti SL: Tb-Fe (100 nm) / Ti (2 nm) / Tb-Fe (100 nm) (fig. 1). Here, we observed a large dipolar coupling that exists between the top and bottom Tb-Fe layers. The Hc of about 0.5 T observed for bare Tb-Fe (200 nm), was found to increase to about 1 T for the Tb-Fe films with Ti SL. Interestingly, Tb-Fe film with SL exhibited a hybrid domain pattern probably due to the bottom Tb-Fe layer. Thus, Ti when used either as UL or SL improved the OOP magnetic properties of Tb-Fe thin films considerably. The fundamental observation and understanding will have a significant impact on perpendicular media which may also useful in other areas of spintronic applications.References: 1. K. Tang, et. al, IEEE transactions on magnetics, Vol. 45 p. 786-792 (2009).

Effect of Ti as an Under and Spacer-layer on the Perpendicular Magnetic Anisotropy of Tb-Fe Thin Film

Atomistic spin models [1] provide a natural bridge between electronic and micromagnetic length scales, where the magnetic properties are described by localised atomic spins within the Heisenberg model. The model parameters can be calculated ab-initio but the macroscopic magnetization dynamics and properties can also be simulated. While powerful, atomistic models are computationally expensive. A particular difficulty is the calculation of long-range dipole-dipole interactions, arising from the stray magnetic fields of localised atomic moments and causes complex magnetisation structures, such as magnetic vortices or Skyrmions. The dipole -dipole interactions are typically solved using a Fast-Fourier-Transform (FFT) [2] or the Fast Multipole Method (FMM) [3]. However, for atomistic simulations the continuum approximation does not accurately reflect the local dipole-dipole interactions at the atomic level and leads to systematic errors in the calculation of the dipole field. A brute force atomistic dipole-dipole approach is infeasible for the time (nanosecond+) and length (100 nm+) scales needed to extract useful data. Here we present modifications to the VAMPIRE software package [1] to include accurate and scalable calculations of dipole-dipole interactions for improved performance and scalability on ARCHER, the UK national supercomputer. We have developed a space-time hierarchical calculation of the dipole field calculation with order NlogN scalability with problem size, a comparison to the brute force tensor approach [4] is shown in Fig. 1. Our novel approach combines exact near-field computation of the dipole field with an M-Level hierarchical representation of the far-field dipole field [5]. The method is hierarchical in time and space, since the far-field time-evolution of the magnetization is much slower than the near-field which changes on the femtosecond timescale. Our improved algorithm enables the inclusion of accurate dipole-dipole interactions in atomistic simulations for large system sizes in the micrometre range for the first time.References: [1] R. F. L. Evans et al, J. Phys.: Condens. Matter 26, 103202 (2014) [2] S. Yuan and H. Bertram, IEEE Trans. Mag. 28, 2031 (1992) [3] G. Brown et al, IEEE Trans. Mag. 40 2146 (2004) [4] G.J. Bowden et al., J. Phys. Condens. Matter 28, 066001 (2016) [5] J.J. Miles and B.K. Middleton, J. Magn. Magn. Mater. 95, 99 (1991)

Scalable space and time hierarchical dipole-dipole interactions in the VAMPIRE code

With the development of electronic low-dimension systems, numerical modeling has become a powerful method for studying and predicting the physical properties of materials. To model dynamic magnetic systems the micromagnetic approach [1] and the methods of GPU acceleration of calculations on the video core of the system [2] are successfully applied. In our work, this approach is used to simulate the effect of nonlinear autoresonant excitation [3, 4] of nanoscale films of yttrium iron garnet (YIG) by a weak external magnetic field. The proposed analytical equations for autoresonance do not take into account a number of important real conditions, for example: attenuation, three-dimensional geometry of the sample, dipole – dipole interaction of spins, surface anisotropy, etc., which does not allow one to directly verify the theoretical model in practice. The aim of the work was to find the conditions for the excitation of autoresonance in a film with some real physical parameters using micromagnetic simulation in the MUMAX3 software. As a planar model, a YIG film with a size of 1600 × 1600 × 160 nm was chosen. The film thickness was taken to satisfy the conditions of a single-mode spin wave excitation and proximity to real planar samples. During simulations the sample was separated into rectangular elements in the form of cells, the interactions between them were calculated by using the Landau-Lifshitz equation. Since the YIG has the minimum level of attenuation of the magnetic subsystem, the damping parameter was excluded at this stage. Then the sample saturated along the axis of easy magnetization by H = 20 kOe was excited by a weak perpendicular pump field h with an amplitude of about 1 Oe. As a result, the stable nonlinear regime of excitation of magnetic oscillations of high magnitude in the film by a chirp rate of the pump field frequency was obtained in in the MUMAX3 simulations for the first time - the autoresonance effect. For the modelled YIG film the optimal chirp rate was a 4.3×10^16 s^(– 2). Finally, the full conditions for the autoresonance excitation of magnetization in the quasi-three-dimensional magnetic film by numerical methods were defined. Supported by the RFBR grant No. 19-32-90014.References: 1 A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen, F. Garcia-Sanchez // AIP Advances. 2014. Vol. 4. P. 107133. 2 V. S. Teplov, V. D. Bessonov. Resource and Mechanics of materials and structures. 2018. V. 6.P. 222—228. 3. S.V. Batalov, A. G. Shagalov. Physics of Metals and Metallography, 2010. Vol 109, N1, P. 3-8 . 4 S,V. Batalov, A. G. Shagalov.. Physics of Metals and Metallography, 2013, Vol 114, N 2, P. 115–119

Micromagnetic modeling magnetization excitation by a resonance

Spherical nanoparticles with hollow structure have received much attention because of their unique magnetic features reflecting their large surface area [1,2]. Recent works suggested the formation of a spin vortex for hollow spherical submicron Fe3O4 particles, whose stability is enhanced with increasing the diameter [3]. In this work, the formation mechanism of spin vortexes have been studied for a hollow submicron sphere using micromagnetic simulations, where possible realization of a double vortex reflecting the hollow structure was revealed. The magnetization reversal and magnetic configurations under magnetic fields along the [100] or [111] axis were studied using OOMMF for a single hollow Fe3O4 spherical particle with the varying diameter (d = 300 - 700 nm) and inner/outer diameter ratio (γ = 0 - 0.75). Electron holography was also performed for 400- nm hollow spheres to directly characterize a magnetic state. Figure 1 shows a typical example of the M-H curves for a hollow sphere (d = 300 nm, γ = 0.5). Before entering a single vortex state (phase IV) typically observed in a solid sphere, two types of vortex structures are formed for a hollow sphere, associated with multi-step magnetization changes. We find that as the magnetic field is reduced from high fields, a clockwise and anti-clockwise vortices are formed at the ends of the sphere (phase II) and then transform into a single vortex with cores displaced by changing a magnetic field (phase III). The formation of such double vortices can be due to reduced Zeeman energy by spins at and around the equatorial plane of the sphere. With increasing d or γ, a field range, where the phases II and III are stabilized, expands for both [100] and [111] field orientations. Magnetic phase images obtained by electron holography revealed no significant stray field around the hollow sphere in zero fields, being consistent with the formation of a spin vortex.References: 1] H. Khurashid et al., Scientific Reports 5, 15054 (2015) 2] Y. Liang et al., Colloids and Surfaces A 530 191-199 (2017) 3] M. Chiba et al., Journal of Magnetism and Magnetic Materials 512 167012 (2020)

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Stress-Induced Modification of Gyration Eigen-Frequencies in Stacked Double-Vortex Structures

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